From Liar Paradox to Incongruent Sets: A Normal Form for Self-Reference

TL;DR

This paper introduces incongruent normal form (INF), a structural method for analyzing self-referential sentences, revealing how local consistency can lead to global inconsistency and providing a quantitative framework for semantic informativeness.

Contribution

The paper presents INF as a novel structural representation for self-reference, along with correctness theorems and a quantitative semantic framework linking incongruence to informativeness and energy.

Findings

INF isolates semantic obstructions caused by self-reference.

Semantic completeness prevents informativeness, while incongruence preserves it.

Incongruence occurs in consistent incomplete theories and relates to semantic energy bounds.

Abstract

We introduce incongruent normal form (INF), a structural representation for self-referential semantic sentences. An INF replaces a self-referential sentence with a finite family of non-self-referential sentences that are individually satisfiable but not jointly satisfiable. This transformation isolates the semantic obstruction created by self-reference while preserving classical semantics locally and is accompanied by correctness theorems characterizing when global inconsistency arises from locally compatible commitments. We then study the role of incongruence as a structural source of semantic informativeness. Using a minimal model-theoretic notion of informativeness-understood as the ability of sentences to distinguish among admissible models-we show that semantic completeness precludes informativeness, while incongruence preserves it. Moreover, incongruence is not confined to paradoxical constructions: any consistent incomplete first-order theory admits finite incongruent families arising from incompatible complete extensions. In this sense, incompleteness manifests structurally as locally realizable but globally incompatible semantic commitments, providing a minimal formal basis for semantic knowledge. Finally, we introduce a quantitative semantic framework. In a canonical finite semantic-state setting, we model semantic commitments as Boolean functions and define a Fourier-analytic notion of semantic energy based on total influence. We derive uncertainty-style bounds relating semantic determinacy, informativeness, and spectral simplicity, and establish a matrix inequality bounding aggregate semantic variance by total semantic energy. These results show quantitatively that semantic informativeness cannot collapse into a single determinate state without unbounded energy cost, identifying incongruence as a fundamental structural and quantitative feature of semantic representation.